# TeX shapes the landscape

Computer games and science fiction movies use procedural landscapes. Why not use TeX to generate some. The approach is:

• Using the diamond-square algorithm for generating the height map
• Using Lua for implementing the algorithm
• Print with pgfplots, that provides a comprehensive interface for 3d plotting
• Let’s make a surface plot mit mesh
• Use a color map: blue for below the water line, green for mountains, white for snow, color gradient according to the height
• Matrix size in powers of two, perhaps adding opacity, starting at zero level for islands, …

That’s an example what we can get:

Another one:

The same with option shader=interp:

The code is:

\documentclass[border=10pt]{standalone}
\usepackage{pgfplots}
\usepackage{luacode}
\begin{luacode*}
function terrain(seed,dimension,options)
-- inner functions come from the Heightmap module
-- Module Copyright (C) 2011 Marc Lepage

local max, random = math.max, math.random

-- Find power of two sufficient for size
local function pot(size)
local pot = 2
while true do
if size &lt;= pot then return pot end
pot = 2*pot
end
end

-- Create a table with 0 to n zero values
local function tcreate(n)
local t = {}
for i = 0, n do t[i] = 0 end
return t
end

-- Square step
-- Sets map[x][y] from square of radius d using height function f
local function square(map, x, y, d, f)
local sum, num = 0, 0
if 0 &lt;= x-d then
if   0 &lt;= y-d   then sum, num = sum + map[x-d][y-d], num + 1 end
if y+d &lt;= map.h then sum, num = sum + map[x-d][y+d], num + 1 end
end
if x+d &lt;= map.w then
if   0 &lt;= y-d   then sum, num = sum + map[x+d][y-d], num + 1 end
if y+d &lt;= map.h then sum, num = sum + map[x+d][y+d], num + 1 end
end
map[x][y] = f(map, x, y, d, sum/num)
end

-- Diamond step
-- Sets map[x][y] from diamond of radius d using height function f
local function diamond(map, x, y, d, f)
local sum, num = 0, 0
if   0 &lt;= x-d   then sum, num = sum + map[x-d][y], num + 1 end
if x+d &lt;= map.w then sum, num = sum + map[x+d][y], num + 1 end
if   0 &lt;= y-d   then sum, num = sum + map[x][y-d], num + 1 end
if y+d &lt;= map.h then sum, num = sum + map[x][y+d], num + 1 end
map[x][y] = f(map, x, y, d, sum/num)
end

-- Diamond square algorithm generates cloud/plasma fractal heightmap
-- http://en.wikipedia.org/wiki/Diamond-square_algorithm
-- Size must be power of two
-- Height function f must look like f(map, x, y, d, h) and return h'
local function diamondsquare(size, f)
-- create map
local map = { w = size, h = size }
for c = 0, size do map[c] = tcreate(size) end
-- seed four corners
local d = size
map[0][0] = f(map, 0, 0, d, 0)
map[0][d] = f(map, 0, d, d, 0)
map[d][0] = f(map, d, 0, d, 0)
map[d][d] = f(map, d, d, d, 0)
d = d/2
-- perform square and diamond steps
while 1 &lt;= d do
for x = d, map.w-1, 2<em>d do
for y = d, map.h-1, 2</em>d do
square(map, x, y, d, f)
end
end
for x = d, map.w-1, 2<em>d do
for y = 0, map.h, 2</em>d do
diamond(map, x, y, d, f)
end
end
for x = 0, map.w, 2<em>d do
for y = d, map.h-1, 2</em>d do
diamond(map, x, y, d, f)
end
end
d = d/2
end
return map
end

-- Default height function
-- d is depth (from size to 1 by powers of two)
-- h is mean height at map[x][y] (from square/diamond of radius d)
-- returns h' which is used to set map[x][y]
function defaultf(map, x, y, d, h)
return h + (random()-0.5)*d
end

-- Create a heightmap using the specified height function (or default)
-- map[x][y] where x from 0 to map.w and y from 0 to map.h
function create(width, height, f)
f = f and f or defaultf
-- make heightmap
local map = diamondsquare(pot(max(width, height)), f)
-- clip heightmap to desired size
for x = 0, map.w do for y = height+1, map.h do map[x][y] = nil end end
for x = width+1, map.w do map[x] = nil end
map.w, map.h = width, height
return map
end

-- Initialize pseudo random number generator with seed, to be able to reproduce
math.randomseed(seed)
map = create(dimension, dimension)
if options ~= [[]] then
.. options .. "
] coordinates{")
else
end
for x = 0, map.w do
for y = 0, map.h do
tex.sprint("("..x..","..y..","..map[x][y]..")")
end
end
tex.sprint("};")
end
\end{luacode*}
\begin{document}
\begin{tikzpicture}
\begin{axis}[colormap={terrain}{color(0cm)=(blue!40!black);
color(1cm)=(blue); color(2cm)=(green!40!black);
color(4cm)=(green!60!white);color(4cm)=(white!95!black);
color(8cm)=(white); color(8cm)=(white)},
hide axis, view = {90}{10}
]
\directlua{terrain(14,128,[[surf,mesh/rows=129,mesh/check=false]])}
\end{axis}
\end{tikzpicture}
\end{document
}

The first image had the seed 10 (first argument of the terrain function) and view={10}{55}.

This was done just for fun, I posted it already in German on TeXwelt.de. Bigger landscapes can be too hard for memory and processing power when using TeX for this, but that’s just today – tomorrow computers will be more capable.